The Derivation and Application of Fundamental Solutions for Unsteady Stokes Equations

2015 ◽  
Vol 31 (6) ◽  
pp. 683-691 ◽  
Author(s):  
C-H. Hsiao ◽  
D.-L. Young
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Fundamental Solutions ◽  
Mass Source ◽  
Reynolds Number Flow ◽  
Singularity Method ◽  
Vector Calculus ◽  
Unsteady Stokes Flow ◽  
Number Flow ◽  
Unsteady Stokes Equations

AbstractIn this paper, two formulations in explicit form to derive the fundamental solutions for two and three dimensional unsteady unbounded Stokes flows due to a mass source and a point force are presented, based on the vector calculus method and also the Hörmander’s method. The mathematical derivation process for the fundamental solutions is detailed. The steady fundamental solutions of Stokes equations can be obtained from the unsteady fundamental solutions by the integral process. As an application, we adopt fundamental solutions: an unsteady Stokeslet and an unsteady potential dipole to validate a simple case that a sphere translates in Stokes or low-Reynolds-number flow by using the singularity method instead by the traditional method which in general limits to the assumption of oscillating flow. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions by making use of the fundamental solutions of the linearly unsteady Stokes equations.

Calculation of Hydrodynamic Forces for Unsteady Stokes Flows by Singularity Integral Equations Based on Fundamental Solutions

2013 ◽  
Vol 30 (2) ◽  
pp. 129-136 ◽  
Author(s):  
C. H. Hsiao ◽  
D. L. Young
Keyword(s):  
Time Domain ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fundamental Solutions ◽  
Hydrodynamic Forces ◽  
Stokes Flows ◽  
Total Strength ◽  
Singularity Method ◽  
Unsteady Stokes Flow ◽  
Convolution Quadrature Method

ABSTRACTThe attractive feature of the singularity method for steady Stokes flows is that the hydrodynamic forces acting on the particle can be calculated by the total strength of distributed singularities. For unsteady Stokes flows, however we have to derive hydrodynamic forces acting on a solid body in terms of the strengths of both unsteady Stokeslets as well as unsteady potential dipoles if mass and force sources are both taken into consideration. Since the hydrodynamic force formulation results in a Volterra integral equation of the first kind, and the strengths are numerically approximated by means of the Lubich convolution quadrature method (CQM) in this study. As far as the numerical solutions of time-domain integral formulations of the unsteady Stokes equations are concerned, this paper requires only the Laplace-domain instead of the time- domain fundamental solutions of the governing equations. The stability and accuracy of the proposed method are verified through some well selected numerical examples. In total we include two examples presenting the accuracy of Lubich CQM, and another two examples for calculating general hydrody-namic forces of a sphere in oscillating or non-oscillating unsteady Stokes flows. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions instead to limit to the oscillating flow assumption.

Vortex Suppression Efficiency of Discontinuous Helicoidal Fins

2007 ◽  
Author(s):  
Antonio Pinto ◽  
Riccardo Broglia ◽  
Elena Ciappi ◽  
Andrea Di Mascio ◽  
Emilio F. Campana ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Computational Time ◽  
Reynolds Number Flow ◽  
Flow Past A Cylinder ◽  
Helical Strakes ◽  
Unsteady Rans ◽  
Number Flow ◽  
Offshore Industry ◽  
Good Trade ◽  
High Reynolds

Vortex-Induced Vibration (VIV) is one of the most demanding areas in the offshore industry, and detailed investigation of the fluid-structure interaction is becoming fundamental for designing new structures able to reduce VIV phenomenon. To carry on such analysis, and get reliable results in term of global coefficients, the correct modelling of turbulence, boundary layer, and separated flows is required. Nonetheless, the more accurate is the simulation, the more costly is the computation. Unsteady RANS simulations provide a good trade-off between numerical accuracy and computational time. This paper presents the analysis of the flow past a cylinder with several three-dimensional helical fins at high Reynolds number. Flow field, vortical structures, and response frequency patterns are analysed. Spectral analysis of data is performed to identify carrier frequencies, deemed to be critical due to the induced vibration of the whole structure. Finally, helical strakes efficiency in reducing the riser vibrations is also addressed, through direct consideration on the carrier shedding frequency.

Three-Dimensional Numerical Simulations of Circular Cylinders Undergoing Two Degree-of-Freedom Vortex-Induced Vibrations

2006 ◽  
Author(s):  
Juan P. Pontaza ◽  
Hamn-Ching Chen
Keyword(s):  
Numerical Simulations ◽  
Three Dimensional ◽  
Operating Conditions ◽  
Degree Of Freedom ◽  
Circular Cylinders ◽  
Time Step ◽  
Reynolds Number Flow ◽  
Number Flow ◽  
High Reynolds ◽  
Two Degree Of Freedom

In an effort to gain a better understanding of the VIV phenomena, we present three-dimensional numerical simulations of VIV of circular cylinders. We consider operating conditions that correspond to high Reynolds number flow, low structural damping, and allow for two-degree of freedom motion. The numerical implementation makes use of overset (Chimera) grids, in a multiple block environment where the workload associated with the blocks is distributed among multiple processors working in parallel. The three-dimensional grids around the cylinder are allowed to undergo arbitrary motions with respect to fixed background grids, eliminating the need for tedious grid regeneration at every time step.

Coherent structures in coflowing jets and wakes

1978 ◽  
Vol 88 (3) ◽  
pp. 451-463 ◽  
Author(s):  
A. E. Perry ◽  
T. T. Lim
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Glass Tube ◽  
Reynolds Numbers ◽  
Reynolds Number Flow ◽  
Naturally Occurring ◽  
Number Flow ◽  
High Reynolds ◽  
Scale Motion

By applying small lateral oscillations to a glass tube from which smoke was issuing, perfectly periodic coflowing jets and wake structures were produced at Reynolds numbers of order 300-1000. These structures remained coherent over long streamwise distances and appeared to be perfectly frozen when viewed under stroboscopic light which was synchronized with the disturbing oscillation. By the use of strobing laser beams, longitudinal sections of the structures were photographed and an account of the geometry of these structures is reported.When the tube was unforced, similar structures occurred but they modulated in scale and frequency, and their orientation was random.A classification of structures is presented and examples are demonstrated in naturally occurring situations such as smoke from a cigarette, the wake behind a three-dimensional blunt body, and the high Reynolds number flow in a plume from a chimney. It is suggested that an examination of these structures may give some insight into the large-scale motion in fully turbulent flow.

LOW REYNOLDS NUMBER FLOW OVER A SQUARE RIB

1997 ◽  
Vol 21 (4) ◽  
pp. 371-387
Author(s):  
D.A Billenness ◽  
N. Djilali ◽  
E. Zeidan
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Separation Bubble ◽  
Three Dimensional ◽  
Recirculation Region ◽  
Number Range ◽  
Reynolds Number Flow ◽  
First Results ◽  
Region Length ◽  
Number Flow

Laminar flow over a square rib placed in a fully developed channel flow is investigated over the Reynolds number range 80-350. The effect of Reynolds number on the flow and the variation of the primary reattachment length with Reynolds number are investigated using flow visualization and laser-Doppler velocimetry. The primary recirculation region length is found to increase in a slightly non-linear fashion with Reynolds number up to Reh = 250, at which point shear layer instabilities first appear downstream of the rib. Increasing the Reynolds number further, first results in continuing growth of the separation bubble, and then for Reh ≳ 300, in the appearance of three dimensional vortices and gradual shortening of the bubble. The measurements are complemented by two- and three-dimensional numerical simulations using a finite volume method with a high-order descretization scheme. These simulations yield excellent agreement with the measured reattachment lengths and velocity profiles over the steady laminar flow régime.

Numerical Simulations of Riser Vortex-Induced Vibrations

2005 ◽  
Author(s):  
Juan P. Pontaza ◽  
Hamn-Ching Chen ◽  
Chia-Rong Chen
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
Vortex Induced Vibrations ◽  
Number Flow ◽  
Low Mass ◽  
High Reynolds ◽  
Low Mass Ratio ◽  
Important Design

Vortex-induced vibrations (VIV) is an important design consideration for marine risers in offshore drilling and production. In an effort to better understand the VIV phenomena, we present numerical simulation results for two-dimensional incompressible flow past freely vibrating multi-cylinder configurations found in offshore engineering. Of interest is the response of the structure for low mass ratio, low damping, and high Reynolds number flow conditions. The governing incompressible Navier-Stokes equations are numerically solved and time-integrated using a local-analytic-based discretization procedure, implemented in conjunction with overset (Chimera) grid capabilities for zonal-based resolution of the flow field.

A spectral boundary element approach to three-dimensional Stokes flow

1995 ◽  
Vol 298 ◽  
pp. 167-192 ◽  
Author(s):  
G. P. Muldowney ◽  
J. J. L. Higdon
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Spectral Element ◽  
Companion Paper ◽  
Porous Membrane ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Reynolds Number Flow ◽  
Periodic Models ◽  
Element Approach

A novel method is introduced for solving the three-dimensional Stokes equations via a spectral element approach to the boundary integral method. The accuracy and convergence of the method are illustrated through applications involving rigid particles, deformable droplets and interacting particles. New physical results are obtained for two applications in low Reynolds number flow: the permeability of periodic models of a porous membrane and the instability of a toroidal droplet subject to non-axisymmetric perturbations. Further applications are described in the companion paper (Higdon & Muldowney 1995).

The interacting boundary-layer flow due to a vortex approaching a cylinder

1997 ◽  
Vol 346 ◽  
pp. 319-343 ◽  
Author(s):  
Z. XIAO ◽  
O. R. BURGGRAF ◽  
A. T. CONLISK
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Boundary Layer Separation ◽  
Two Dimensions ◽  
Unsteady Boundary Layer ◽  
Reynolds Number Flow ◽  
Boundary Layer Equations ◽  
Number Flow ◽  
Layer Flow ◽  
High Reynolds

In this paper the solution to the three-dimensional and unsteady interacting boundary-layer equations for a vortex approaching a cylinder is calculated. The flow is three-dimensional and unsteady. The purpose of this paper is to enhance the understanding of the structure in three-dimensional unsteady boundary-layer separation commonly observed in a high-Reynolds-number flow. The short length scales associated with the boundary-layer eruption process are resolved through an efficient and effective moving adaptive grid procedure. The results of this work suggest that like its two-dimensional counterpart, the three-dimensional unsteady interacting boundary layer also terminates in a singularity at a finite time. Furthermore, the numerical calculations confirm the theoretical analysis of the singular structure in two dimensions for the interacting boundary layer due to Smith (1988).

The general solution for an ellipsoid in low-Reynolds-number flow

1987 ◽  
Vol 178 ◽  
pp. 535-547 ◽  
Author(s):  
Sangtae Kim ◽  
P. V. Arunachalam
Keyword(s):  
Reynolds Number ◽  
General Solution ◽  
Low Reynolds Number ◽  
Multipole Moments ◽  
Low Reynolds Number Flow ◽  
Reynolds Number Flow ◽  
Singularity Method ◽  
Number Flow ◽  
Low Reynolds ◽  
Singularity Distribution

The general solution for low-Reynolds-number flow about an ellipsoid is derived by the singularity method and by representation in ellipsoidal harmonics. It is shown that, as in potential flow, the focal ellipse is the image system for the ellipsoid. A simple transformation which resembles a step in the derivation of the Dirichlet potential is introduced and its implications are explored. This transformation converts the velocity representation for an nth-order ambient field into that for the (n + 1)th-order field. The method furnishes an explanation for the invariance of the domain of the singularity distribution (the focal ellipse) with respect to the ambient field. Faxén relations for all multipole moments for arbitrary Stokes flow are derived in both integral and symbolic operator forms.

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